ABCF->ab-angle a

Percentage Accurate: 18.8% → 61.4%

Time: 14.2s

Alternatives: 15

Speedup: 16.9×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Initial Program: 18.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) + sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) + math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) + sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Alternative 1: 61.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B\_m\right)\\ t_1 := C \cdot \left(A \cdot 4\right)\\ t_2 := t\_1 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B\_m}^{2}}\right) \cdot \left(\left(F \cdot \left({B\_m}^{2} - t\_1\right)\right) \cdot 2\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right) \cdot 2}\right) \cdot \sqrt{\left(C + A\right) + t\_0}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right) \cdot F} \cdot \sqrt{t\_4 \cdot 2}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot t\_4}}{t\_4} \cdot \frac{\sqrt{\left(t\_0 + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot (- A C) B_m))
        (t_1 (* C (* A 4.0)))
        (t_2 (- t_1 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt
           (*
            (+ (+ C A) (sqrt (+ (pow (- A C) 2.0) (pow B_m 2.0))))
            (* (* F (- (pow B_m 2.0) t_1)) 2.0)))
          t_2))
        (t_4 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= t_3 -5e-195)
     (/
      (*
       (* (sqrt F) (sqrt (* (fma (* C A) -4.0 (* B_m B_m)) 2.0)))
       (sqrt (+ (+ C A) t_0)))
      t_2)
     (if (<= t_3 0.0)
       (/
        (*
         (sqrt (* (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C) F))
         (sqrt (* t_4 2.0)))
        t_2)
       (if (<= t_3 INFINITY)
         (* (/ (sqrt (* (* F 2.0) t_4)) t_4) (/ (sqrt (+ (+ t_0 A) C)) -1.0))
         (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot((A - C), B_m);
	double t_1 = C * (A * 4.0);
	double t_2 = t_1 - pow(B_m, 2.0);
	double t_3 = sqrt((((C + A) + sqrt((pow((A - C), 2.0) + pow(B_m, 2.0)))) * ((F * (pow(B_m, 2.0) - t_1)) * 2.0))) / t_2;
	double t_4 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (t_3 <= -5e-195) {
		tmp = ((sqrt(F) * sqrt((fma((C * A), -4.0, (B_m * B_m)) * 2.0))) * sqrt(((C + A) + t_0))) / t_2;
	} else if (t_3 <= 0.0) {
		tmp = (sqrt(((((((B_m * B_m) / A) * -0.5) + C) + C) * F)) * sqrt((t_4 * 2.0))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt(((F * 2.0) * t_4)) / t_4) * (sqrt(((t_0 + A) + C)) / -1.0);
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = hypot(Float64(A - C), B_m)
	t_1 = Float64(C * Float64(A * 4.0))
	t_2 = Float64(t_1 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(Float64(C + A) + sqrt(Float64((Float64(A - C) ^ 2.0) + (B_m ^ 2.0)))) * Float64(Float64(F * Float64((B_m ^ 2.0) - t_1)) * 2.0))) / t_2)
	t_4 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (t_3 <= -5e-195)
		tmp = Float64(Float64(Float64(sqrt(F) * sqrt(Float64(fma(Float64(C * A), -4.0, Float64(B_m * B_m)) * 2.0))) * sqrt(Float64(Float64(C + A) + t_0))) / t_2);
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C) * F)) * sqrt(Float64(t_4 * 2.0))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * t_4)) / t_4) * Float64(sqrt(Float64(Float64(t_0 + A) + C)) / -1.0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(N[(C + A), $MachinePrecision] + N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-195], N[(N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(C + A), $MachinePrecision] + t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$4 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * t$95$4), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision] * N[(N[Sqrt[N[(N[(t$95$0 + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision] / -1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000009e-195

   1. Initial program 46.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*l* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 67.3%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   5. Step-by-step derivation

      1. lift-neg.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. distribute-lft-neg-in N/A

         \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-sqrt.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot \color{blue}{\sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. pow1/2 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot \color{blue}{{\left(F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot {\color{blue}{\left(F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)\right)}}^{\frac{1}{2}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. unpow-prod-down N/A

         \[\leadsto \frac{\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot \color{blue}{\left({F}^{\frac{1}{2}} \cdot {\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}^{\frac{1}{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. associate-*r* N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}\right)\right) \cdot {F}^{\frac{1}{2}}\right) \cdot {\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}^{\frac{1}{2}}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   6. Applied rewrites 75.2%

      \[\leadsto \frac{\color{blue}{\left(\left(-\sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{F}\right) \cdot \sqrt{\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -5.00000000000000009e-195 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 3.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*l* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 17.5%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. lower-*.f64 27.5

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   7. Applied rewrites 27.5%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 27.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 94.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1} \cdot \frac{\sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 17.3

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 17.3%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 27.7%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 27.8%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 50.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq -5 \cdot 10^{-195}:\\ \;\;\;\;\frac{\left(\sqrt{F} \cdot \sqrt{\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2}\right) \cdot \sqrt{\left(C + A\right) + \mathsf{hypot}\left(A - C, B\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{C \cdot \left(A \cdot 4\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(\left(C + A\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right) \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right) \cdot 2\right)}}{C \cdot \left(A \cdot 4\right) - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \frac{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e-120)
     (/
      -1.0
      (/
       t_0
       (sqrt (* (* (* F 2.0) t_0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))))
     (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-120) {
		tmp = -1.0 / (t_0 / sqrt((((F * 2.0) * t_0) * (((((B_m * B_m) / A) * -0.5) + C) + C))));
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-120)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-120], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999996e-120

   1. Initial program 22.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 31.1%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      5. lower-*.f64 24.5

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   7. Applied rewrites 24.5%

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   if 1.99999999999999996e-120 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 20.5

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 20.5%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.1%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 30.2%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-120}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 51.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= (pow B_m 2.0) 2e+123)
     (/
      -1.0
      (/
       t_0
       (sqrt (* (* (* F 2.0) t_0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))))
     (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+123) {
		tmp = -1.0 / (t_0 / sqrt((((F * 2.0) * t_0) * (((((B_m * B_m) / A) * -0.5) + C) + C))));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+123)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+123], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999996e123

   1. Initial program 25.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 34.4%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      5. lower-*.f64 22.3

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   7. Applied rewrites 22.3%

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   if 1.99999999999999996e123 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 26.4

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 26.4%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 39.8%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 39.9%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{2 \cdot \left(B + C\right)}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 36.5%

       \[\leadsto \frac{\sqrt{2 \cdot \left(C + B\right)}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+123}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ \mathbf{if}\;B\_m \leq 1.7 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(t\_0 \cdot 2\right)\right) \cdot F}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 58000000000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m))) (t_1 (- t_0)))
   (if (<= B_m 1.7e-225)
     (/ (sqrt (* (* (* C 2.0) (* t_0 2.0)) F)) t_1)
     (if (<= B_m 58000000000.0)
       (/
        (*
         (sqrt (* (* F 2.0) (fma -4.0 (* C A) (* B_m B_m))))
         (sqrt (+ (+ (hypot (- A C) B_m) A) C)))
        t_1)
       (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = -t_0;
	double tmp;
	if (B_m <= 1.7e-225) {
		tmp = sqrt((((C * 2.0) * (t_0 * 2.0)) * F)) / t_1;
	} else if (B_m <= 58000000000.0) {
		tmp = (sqrt(((F * 2.0) * fma(-4.0, (C * A), (B_m * B_m)))) * sqrt(((hypot((A - C), B_m) + A) + C))) / t_1;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (B_m <= 1.7e-225)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * Float64(t_0 * 2.0)) * F)) / t_1);
	elseif (B_m <= 58000000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(F * 2.0) * fma(-4.0, Float64(C * A), Float64(B_m * B_m)))) * sqrt(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[B$95$m, 1.7e-225], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 58000000000.0], N[(N[(N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.7e-225

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \color{blue}{2}\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 12.3

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 12.3%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.7e-225 < B < 5.8e10

   1. Initial program 30.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   8. Applied rewrites 59.5%

      \[\leadsto \frac{\color{blue}{\sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C} \cdot \sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5.8e10 < B

   1. Initial program 13.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 48.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 74.8%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.7 \cdot 10^{-225}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 58000000000:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)} \cdot \sqrt{\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 55.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ t_1 := -t\_0\\ \mathbf{if}\;B\_m \leq 1.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(t\_0 \cdot 2\right)\right) \cdot F}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 58000000000:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot 2} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right) \cdot F}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m))) (t_1 (- t_0)))
   (if (<= B_m 1.2e-224)
     (/ (sqrt (* (* (* C 2.0) (* t_0 2.0)) F)) t_1)
     (if (<= B_m 58000000000.0)
       (/
        (*
         (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) 2.0))
         (sqrt (* (fma -4.0 (* C A) (* B_m B_m)) F)))
        t_1)
       (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double t_1 = -t_0;
	double tmp;
	if (B_m <= 1.2e-224) {
		tmp = sqrt((((C * 2.0) * (t_0 * 2.0)) * F)) / t_1;
	} else if (B_m <= 58000000000.0) {
		tmp = (sqrt((((hypot((A - C), B_m) + A) + C) * 2.0)) * sqrt((fma(-4.0, (C * A), (B_m * B_m)) * F))) / t_1;
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if (B_m <= 1.2e-224)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * Float64(t_0 * 2.0)) * F)) / t_1);
	elseif (B_m <= 58000000000.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * 2.0)) * sqrt(Float64(fma(-4.0, Float64(C * A), Float64(B_m * B_m)) * F))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[B$95$m, 1.2e-224], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 58000000000.0], N[(N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.20000000000000007e-224

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 23.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 18.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \color{blue}{2}\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 12.3

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 12.3%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 1.20000000000000007e-224 < B < 5.8e10

   1. Initial program 30.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 41.8%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 31.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   8. Applied rewrites 59.3%

      \[\leadsto \frac{\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F} \cdot \sqrt{2 \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 5.8e10 < B

   1. Initial program 13.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 48.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 48.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 74.5%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 74.8%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-224}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{elif}\;B \leq 58000000000:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot 2} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 57.0% accurate, 2.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(-4, C \cdot A, B\_m \cdot B\_m\right)\\ \mathbf{if}\;B\_m \leq 7.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(\left(F \cdot 2\right) \cdot t\_0\right) \cdot \left(\left(\frac{B\_m \cdot B\_m}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{elif}\;B\_m \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\_m\right) + A\right) + C\right) \cdot F} \cdot \sqrt{t\_0 \cdot 2}}{-\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\_m\right) + C\right) \cdot 2}}{-B\_m} \cdot \sqrt{F}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma -4.0 (* C A) (* B_m B_m))))
   (if (<= B_m 7.2e-109)
     (/
      -1.0
      (/
       t_0
       (sqrt (* (* (* F 2.0) t_0) (+ (+ (* (/ (* B_m B_m) A) -0.5) C) C)))))
     (if (<= B_m 1.8)
       (/
        (* (sqrt (* (+ (+ (hypot (- A C) B_m) A) C) F)) (sqrt (* t_0 2.0)))
        (- (fma B_m B_m (* -4.0 (* C A)))))
       (* (/ (sqrt (* (+ (hypot C B_m) C) 2.0)) (- B_m)) (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(-4.0, (C * A), (B_m * B_m));
	double tmp;
	if (B_m <= 7.2e-109) {
		tmp = -1.0 / (t_0 / sqrt((((F * 2.0) * t_0) * (((((B_m * B_m) / A) * -0.5) + C) + C))));
	} else if (B_m <= 1.8) {
		tmp = (sqrt((((hypot((A - C), B_m) + A) + C) * F)) * sqrt((t_0 * 2.0))) / -fma(B_m, B_m, (-4.0 * (C * A)));
	} else {
		tmp = (sqrt(((hypot(C, B_m) + C) * 2.0)) / -B_m) * sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(-4.0, Float64(C * A), Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 7.2e-109)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(Float64(F * 2.0) * t_0) * Float64(Float64(Float64(Float64(Float64(B_m * B_m) / A) * -0.5) + C) + C)))));
	elseif (B_m <= 1.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(hypot(Float64(A - C), B_m) + A) + C) * F)) * sqrt(Float64(t_0 * 2.0))) / Float64(-fma(B_m, B_m, Float64(-4.0 * Float64(C * A)))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(hypot(C, B_m) + C) * 2.0)) / Float64(-B_m)) * sqrt(F));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(-4.0 * N[(C * A), $MachinePrecision] + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7.2e-109], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(N[(F * 2.0), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] / A), $MachinePrecision] * -0.5), $MachinePrecision] + C), $MachinePrecision] + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.8], N[(N[(N[Sqrt[N[(N[(N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + A), $MachinePrecision] + C), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision] + C), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 7.2000000000000001e-109

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   5. Taylor expanded in A around -inf

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]
   6. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + \frac{-1}{2} \cdot \frac{{B}^{2}}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \color{blue}{\frac{-1}{2} \cdot \frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      3. lower-/.f64 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \color{blue}{\frac{{B}^{2}}{A}}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      4. unpow2 N/A

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + \frac{-1}{2} \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

      5. lower-*.f64 17.9

         \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(C + -0.5 \cdot \frac{\color{blue}{B \cdot B}}{A}\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   7. Applied rewrites 17.9%

      \[\leadsto \frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\color{blue}{\left(C + -0.5 \cdot \frac{B \cdot B}{A}\right)} + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}} \]

   if 7.2000000000000001e-109 < B < 1.80000000000000004

   1. Initial program 23.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \color{blue}{\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)}\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. associate-*r* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot F\right)} \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. associate-*l* N/A

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. sqrt-prod N/A

         \[\leadsto \frac{-\color{blue}{\sqrt{2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. pow1/2 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{-\color{blue}{{\left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}^{\frac{1}{2}} \cdot \sqrt{F \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied rewrites 38.9%

      \[\leadsto \frac{-\color{blue}{\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   5. Taylor expanded in B around 0

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\color{blue}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
   6. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\color{blue}{{B}^{2} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\color{blue}{B \cdot B} + \left(\mathsf{neg}\left(4\right)\right) \cdot \left(A \cdot C\right)} \]

      3. metadata-eval N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{B \cdot B + \color{blue}{-4} \cdot \left(A \cdot C\right)} \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(B, B, \color{blue}{-4 \cdot \left(A \cdot C\right)}\right)} \]

      6. lower-*.f64 38.9

         \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \color{blue}{\left(A \cdot C\right)}\right)} \]

   7. Applied rewrites 38.9%

      \[\leadsto \frac{-\sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2} \cdot \sqrt{F \cdot \left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right)}}{\color{blue}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}} \]

   if 1.80000000000000004 < B

   1. Initial program 13.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 47.3

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 47.3%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.1%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 72.3%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(F \cdot 2\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right) \cdot \left(\left(\frac{B \cdot B}{A} \cdot -0.5 + C\right) + C\right)}}}\\ \mathbf{elif}\;B \leq 1.8:\\ \;\;\;\;\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot F} \cdot \sqrt{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right) \cdot 2}}{-\mathsf{fma}\left(B, B, -4 \cdot \left(C \cdot A\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot 2}}{-B} \cdot \sqrt{F}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 50.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(t\_0 \cdot 2\right)\right) \cdot F}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma (* C A) -4.0 (* B_m B_m))))
   (if (<= (pow B_m 2.0) 1e+23)
     (/ (sqrt (* (* (* C 2.0) (* t_0 2.0)) F)) (- t_0))
     (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma((C * A), -4.0, (B_m * B_m));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+23) {
		tmp = sqrt((((C * 2.0) * (t_0 * 2.0)) * F)) / -t_0;
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(Float64(C * A), -4.0, Float64(B_m * B_m))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+23)
		tmp = Float64(sqrt(Float64(Float64(Float64(C * 2.0) * Float64(t_0 * 2.0)) * F)) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+23], N[(N[Sqrt[N[(N[(N[(C * 2.0), $MachinePrecision] * N[(t$95$0 * 2.0), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999992e22

   1. Initial program 25.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 35.2%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 27.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in C around inf

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot \left(2 + \left(-1 \cdot \frac{A}{C} + \frac{A}{C}\right)\right)\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. distribute-lft1-in N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{\left(-1 + 1\right) \cdot \frac{A}{C}}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0} \cdot \frac{A}{C}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. mul0-lft N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \left(2 + \color{blue}{0}\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\left(\left(C \cdot \color{blue}{2}\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 19.3

         \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 19.3%

      \[\leadsto \frac{\sqrt{\left(\color{blue}{\left(C \cdot 2\right)} \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 9.9999999999999992e22 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 23.6

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 23.6%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 35.7%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 35.8%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{2 \cdot \left(B + C\right)}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 31.3%

       \[\leadsto \frac{\sqrt{2 \cdot \left(C + B\right)}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(\left(C \cdot 2\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 45.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot -16\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-141)
   (/ (sqrt (* (* (* (* C C) A) -16.0) F)) (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-141) {
		tmp = sqrt(((((C * C) * A) * -16.0) * F)) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-141)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * A) * -16.0) * F)) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-141], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-141

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot {C}^{2}\right)\right)} \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot {C}^{2}\right)\right)} \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(-16 \cdot \color{blue}{\left(A \cdot {C}^{2}\right)}\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. unpow2 N/A

         \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. lower-*.f64 16.8

         \[\leadsto \frac{\sqrt{\left(-16 \cdot \left(A \cdot \color{blue}{\left(C \cdot C\right)}\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 16.8%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-16 \cdot \left(A \cdot \left(C \cdot C\right)\right)\right)} \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 4.9999999999999999e-141 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 20.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.2%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 30.3%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{2 \cdot \left(B + C\right)}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.0%

       \[\leadsto \frac{\sqrt{2 \cdot \left(C + B\right)}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot A\right) \cdot -16\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 45.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(C \cdot A, -4, B\_m \cdot B\_m\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-141)
   (/ (sqrt (* (* (* (* C C) F) A) -16.0)) (- (fma (* C A) -4.0 (* B_m B_m))))
   (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-141) {
		tmp = sqrt(((((C * C) * F) * A) * -16.0)) / -fma((C * A), -4.0, (B_m * B_m));
	} else {
		tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-141)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(C * C) * F) * A) * -16.0)) / Float64(-fma(Float64(C * A), -4.0, Float64(B_m * B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-141], N[(N[Sqrt[N[(N[(N[(N[(C * C), $MachinePrecision] * F), $MachinePrecision] * A), $MachinePrecision] * -16.0), $MachinePrecision]], $MachinePrecision] / (-N[(N[(C * A), $MachinePrecision] * -4.0 + N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-141

   1. Initial program 21.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   4. Applied rewrites 30.7%

      \[\leadsto \color{blue}{\frac{-1}{\frac{\mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)}{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + A\right) + C\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(-4, C \cdot A, B \cdot B\right)\right)}}}} \]

   6. Applied rewrites 23.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(C + A\right)\right) \cdot \left(\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right) \cdot 2\right)\right) \cdot F}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]
   8. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \color{blue}{\left(A \cdot \left({C}^{2} \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \color{blue}{\left({C}^{2} \cdot F\right)}\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      4. unpow2 N/A

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

      5. lower-*.f64 17.3

         \[\leadsto \frac{\sqrt{-16 \cdot \left(A \cdot \left(\color{blue}{\left(C \cdot C\right)} \cdot F\right)\right)}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   9. Applied rewrites 17.3%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left(A \cdot \left(\left(C \cdot C\right) \cdot F\right)\right)}}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)} \]

   if 4.9999999999999999e-141 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 18.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      4. lower-neg.f64 N/A

         \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      5. lower-/.f64 N/A

         \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      6. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

      7. lower-sqrt.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      8. *-commutative N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

      10. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      11. lower-+.f64 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

      12. +-commutative N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

      13. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

      14. unpow2 N/A

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

      15. lower-hypot.f64 20.8

          \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

   5. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
   6. Step-by-step derivation

   7. Applied rewrites 30.2%

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 30.3%

      \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]

   10. Taylor expanded in C around 0

       \[\leadsto \frac{\sqrt{2 \cdot \left(B + C\right)}}{-B} \cdot \sqrt{F} \]
   11. Step-by-step derivation

   12. Applied rewrites 26.0%

       \[\leadsto \frac{\sqrt{2 \cdot \left(C + B\right)}}{-B} \cdot \sqrt{F} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-141}:\\ \;\;\;\;\frac{\sqrt{\left(\left(\left(C \cdot C\right) \cdot F\right) \cdot A\right) \cdot -16}}{-\mathsf{fma}\left(C \cdot A, -4, B \cdot B\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 36.6% accurate, 10.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{\left(C + B\_m\right) \cdot 2}}{B\_m} \cdot \left(-\sqrt{F}\right) \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (/ (sqrt (* (+ C B_m) 2.0)) B_m) (- (sqrt F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt(((c + b_m) * 2.0d0)) / b_m) * -sqrt(f)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (Math.sqrt(((C + B_m) * 2.0)) / B_m) * -Math.sqrt(F);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return (math.sqrt(((C + B_m) * 2.0)) / B_m) * -math.sqrt(F)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(sqrt(Float64(Float64(C + B_m) * 2.0)) / B_m) * Float64(-sqrt(F)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (sqrt(((C + B_m) * 2.0)) / B_m) * -sqrt(F);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(N[Sqrt[N[(N[(C + B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   2. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{\sqrt{2}}{B}\right)\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   4. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right)} \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(-\color{blue}{\frac{\sqrt{2}}{B}}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\frac{\color{blue}{\sqrt{2}}}{B}\right) \cdot \sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   8. *-commutative N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

   9. lower-*.f64 N/A

      \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(C + \sqrt{{B}^{2} + {C}^{2}}\right) \cdot F}} \]

   10. +-commutative N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

   11. lower-+.f64 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\color{blue}{\left(\sqrt{{B}^{2} + {C}^{2}} + C\right)} \cdot F} \]

   12. +-commutative N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{{C}^{2} + {B}^{2}}} + C\right) \cdot F} \]

   13. unpow2 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{\color{blue}{C \cdot C} + {B}^{2}} + C\right) \cdot F} \]

   14. unpow2 N/A

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\sqrt{C \cdot C + \color{blue}{B \cdot B}} + C\right) \cdot F} \]

   15. lower-hypot.f64 14.1

       \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\color{blue}{\mathsf{hypot}\left(C, B\right)} + C\right) \cdot F} \]

5. Applied rewrites 14.1%

   \[\leadsto \color{blue}{\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{\left(\mathsf{hypot}\left(C, B\right) + C\right) \cdot F}} \]
6. Step-by-step derivation

7. Applied rewrites 19.8%

   \[\leadsto \left(-\frac{\sqrt{2}}{B}\right) \cdot \left(\sqrt{\mathsf{hypot}\left(C, B\right) + C} \cdot \color{blue}{\sqrt{F}}\right) \]
8. Step-by-step derivation

9. Applied rewrites 19.9%

   \[\leadsto \frac{\sqrt{2 \cdot \left(\mathsf{hypot}\left(C, B\right) + C\right)}}{-B} \cdot \color{blue}{\sqrt{F}} \]

10. Taylor expanded in C around 0

    \[\leadsto \frac{\sqrt{2 \cdot \left(B + C\right)}}{-B} \cdot \sqrt{F} \]
11. Step-by-step derivation

12. Applied rewrites 16.3%

    \[\leadsto \frac{\sqrt{2 \cdot \left(C + B\right)}}{-B} \cdot \sqrt{F} \]

13. Final simplification 16.3%

    \[\leadsto \frac{\sqrt{\left(C + B\right) \cdot 2}}{B} \cdot \left(-\sqrt{F}\right) \]
14. Add Preprocessing

Alternative 11: 36.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{F \cdot 2}}{-\sqrt{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (sqrt (* F 2.0)) (- (sqrt B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * 2.0)) / -sqrt(B_m);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * 2.0d0)) / -sqrt(b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * 2.0)) / -Math.sqrt(B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return math.sqrt((F * 2.0)) / -math.sqrt(B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(F * 2.0)) / Float64(-sqrt(B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * 2.0)) / -sqrt(B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * 2.0), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 12.8

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 12.8%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 16.7%

   \[\leadsto -\frac{\sqrt{F \cdot 2}}{\sqrt{B}} \]

10. Final simplification 16.7%

    \[\leadsto \frac{\sqrt{F \cdot 2}}{-\sqrt{B}} \]
11. Add Preprocessing

Alternative 12: 36.1% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (/ (- (sqrt F)) (sqrt (* 0.5 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(F) / sqrt((0.5 * B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) / sqrt((0.5d0 * b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) / Math.sqrt((0.5 * B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) / math.sqrt((0.5 * B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) / sqrt(Float64(0.5 * B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) / sqrt((0.5 * B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[N[(0.5 * B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 12.8

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 12.8%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 12.8%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
10. Step-by-step derivation

11. Applied rewrites 16.7%

    \[\leadsto -\frac{\sqrt{F}}{\sqrt{B \cdot 0.5}} \]

12. Final simplification 16.7%

    \[\leadsto \frac{-\sqrt{F}}{\sqrt{0.5 \cdot B}} \]
13. Add Preprocessing

Alternative 13: 36.0% accurate, 12.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (- (sqrt F)) (sqrt (/ 2.0 B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(F) * sqrt((2.0 / B_m));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(f) * sqrt((2.0d0 / b_m))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(F) * Math.sqrt((2.0 / B_m));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(F) * math.sqrt((2.0 / B_m))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(Float64(-sqrt(F)) * sqrt(Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(F) * sqrt((2.0 / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[((-N[Sqrt[F], $MachinePrecision]) * N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 12.8

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 12.8%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 16.6%

   \[\leadsto -\sqrt{F} \cdot \sqrt{\frac{2}{B}} \]

10. Final simplification 16.6%

    \[\leadsto \left(-\sqrt{F}\right) \cdot \sqrt{\frac{2}{B}} \]
11. Add Preprocessing

Alternative 14: 27.5% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{F}{B\_m} \cdot 2} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* (/ F B_m) 2.0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(((F / B_m) * 2.0));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((f / b_m) * 2.0d0))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(((F / B_m) * 2.0));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(((F / B_m) * 2.0))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(F / B_m) * 2.0)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(((F / B_m) * 2.0));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(F / B$95$m), $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 12.8

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 12.8%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Add Preprocessing

Alternative 15: 27.5% accurate, 16.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{\frac{2}{B\_m} \cdot F} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* (/ 2.0 B_m) F))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt(((2.0 / B_m) * F));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 / b_m) * f))
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.sqrt(((2.0 / B_m) * F));
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt(((2.0 / B_m) * F))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(Float64(2.0 / B_m) * F)))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(((2.0 / B_m) * F));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(2.0 / B$95$m), $MachinePrecision] * F), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 19.7%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} \]

   2. *-commutative N/A

      \[\leadsto \mathsf{neg}\left(\color{blue}{\sqrt{2} \cdot \sqrt{\frac{F}{B}}}\right) \]

   3. distribute-lft-neg-in N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   4. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\sqrt{2}\right)\right) \cdot \sqrt{\frac{F}{B}}} \]

   5. lower-neg.f64 N/A

      \[\leadsto \color{blue}{\left(-\sqrt{2}\right)} \cdot \sqrt{\frac{F}{B}} \]

   6. lower-sqrt.f64 N/A

      \[\leadsto \left(-\color{blue}{\sqrt{2}}\right) \cdot \sqrt{\frac{F}{B}} \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left(-\sqrt{2}\right) \cdot \color{blue}{\sqrt{\frac{F}{B}}} \]

   8. lower-/.f64 12.8

      \[\leadsto \left(-\sqrt{2}\right) \cdot \sqrt{\color{blue}{\frac{F}{B}}} \]

5. Applied rewrites 12.8%

   \[\leadsto \color{blue}{\left(-\sqrt{2}\right) \cdot \sqrt{\frac{F}{B}}} \]
6. Step-by-step derivation

7. Applied rewrites 12.8%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

9. Applied rewrites 12.8%

   \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]

10. Final simplification 12.8%

    \[\leadsto -\sqrt{\frac{2}{B} \cdot F} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024331 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))